Nonsimplicial toric Nullstellensatz and stacky GKZ theory

Technical Summary: Nonsimplicial Toric Nullstellensatz and Stacky GKZ Theory

Problem Statement
Classical toric geometry establishes a robust dictionary between the geometry of simplicial toric varieties and multigraded commutative algebra, primarily through the Cox ring $S$ and the irrelevant ideal $B$. This framework, pioneered by Cox, generalizes Hilbert's Nullstellensatz to simplicial toric varieties, providing a bijective correspondence between closed subvarieties and radical homogeneous ideals contained in $B$. However, this correspondence fails for nonsimplicial toric varieties. In the nonsimplicial case, the Cox ring $S$ (graded by the full divisor class group $\text{Cl}(X)$) and the irrelevant ideal $B$ do not suffice to classify subvarieties or subschemes uniquely. Specifically, distinct radical homogeneous ideals in $S$ may correspond to the same subvariety, and the classification of coherent sheaves becomes ambiguous because the variety $X$ is no longer a geometric quotient of an affine scheme by a torus. Furthermore, while toric varieties naturally arise in the context of the Gelfand–Kapranov–Zelevinsky (GKZ) theory of birational geometry, the functoriality of the association between cones in the GKZ fan and toric varieties breaks down when attempting to lift these constructions to their associated canonical stacks.

Methodology
The authors address these deficiencies by shifting the grading of the Cox ring from the full divisor class group $\text{Cl}(X)$ to the subgroup of Q-Cartier divisors, denoted $\text{Q-Car}(X)$.

1. The Q-Cartier Cox Ring ($\check{S}$): The paper defines a new ring $\check{S} = \bigoplus_{d \in \text{Q-Car}(X)} H^0(X, \mathcal{O}_X(d))$, which is a subring of the standard Cox ring $S$. The corresponding irrelevant ideal is $\check{B} = B \cap \check{S}$. This construction restricts attention to degrees where some integer multiple is a Cartier divisor.
2. The Q-Cartier Stack ($X^{\text{Q-Car}}$): The authors associate to $X$ a toric Deligne–Mumford (DM) stack defined as the global quotient $X^{\text{Q-Car}} := [\text{Spec}(\check{S}) - V(\check{B}) / \check{G}]$, where $\check{G} = \text{Hom}_{\mathbb{Z}}(\text{Q-Car}(X), \mathbb{C}^*)$.
3. Toric Stack Data: To provide a combinatorial description of $X^{\text{Q-Car}}$, the authors construct a specific toric stack datum $(\tilde{N}, N, \tilde{\Sigma}, \beta)$. This involves defining a lattice $\tilde{N}$ as a quotient of the free abelian group on the rays by a sublattice encoding the linear dependencies of rays in nonsimplicial cones. The map $\beta$ induces a combinatorial equivalence between the fan $\tilde{\Sigma}$ and the original fan $\Sigma$.
4. Stacky GKZ Theory: The methodology is extended to the GKZ setting. For each cone $\Gamma$ in the GKZ fan $\Sigma_{\text{GKZ}}$, the authors define a toric stack $X_\Gamma$ using a modified stack datum that incorporates "extraneous rays" (weights not in the current fan or irrelevant ideal). This construction ensures that the assignment $\Gamma \mapsto X_\Gamma$ is functorial.

Key Contributions and Results

• Toric Ideal-Variety Correspondence (Theorem A): The paper proves that for any semiprojective toric variety $X$ (simplicial or not), there is a bijective correspondence between closed subvarieties of $X$ and radical homogeneous ideals $I \subseteq \check{B} \subseteq \check{S}$. This generalizes Cox's result to the nonsimplicial case by utilizing the Q-Cartier Cox ring. The proof is provided both via direct algebraic geometry (using geometric quotients) and via stack theory.
• Classification of Sheaves and Subschemes (Theorem B): The authors establish that $X^{\text{Q-Car}}$ is a toric DM stack with coarse moduli space $X$. Crucially, $X^{\text{Q-Car}}$ is universal for the property that the pullback of any Q-Cartier divisor is Cartier.
  • The category of quasi-coherent sheaves on $X^{\text{Q-Car}}$ is equivalent to the category of $\text{Q-Car}(X)$-graded $\check{S}$-modules modulo $\check{B}$-torsion.
  • The stack $X^{\text{Q-Car}}$ is smooth if and only if $X$ is simplicial.
  • This framework resolves the ambiguity in classifying subschemes: distinct ideals in $\check{S}$ correspond to distinct subschemes of the stack, which map to the same subscheme of $X$ only if they differ by $\check{B}$-torsion.
• Comparison with Other Rings: The paper clarifies the relationship between three rings: the standard Cox ring $S$ (Weil divisors), the Q-Cartier Cox ring $\check{S}$, and the Picard ring $S_{\text{Pic}}$ (Cartier divisors). It shows that $S_{\text{Pic}} \subseteq \check{S} \subseteq S$ and that the associated stacks form a sequence of morphisms $X_{\text{can}} \to X^{\text{Q-Car}} \to X_{\text{Pic}} \to X$, where $X_{\text{Pic}}$ is isomorphic to $X$.
• Stacky GKZ Theory (Theorem C): The paper constructs a functorial diagram of toric DM stacks $F: \Sigma_{\text{GKZ}} \to \{\text{DM stacks}\}$ that lifts the standard functoriality of GKZ theory for varieties.
  • For each cone $\Gamma \in \Sigma_{\text{GKZ}}$, there exists a stack $X_\Gamma$ with coarse moduli space $X_\Gamma$.
  • The assignment is functorial with respect to inclusions of cones, restoring the functoriality lost when using canonical stacks.
  • $X_\Gamma$ lies between the variety $X_\Gamma$ and the Q-Cartier stack $X_{\Gamma, \text{Q-Car}}$ via finite maps. When $\Gamma$ lies in the moving cone, $X_\Gamma = X_{\Gamma, \text{Q-Car}}$.
  • This construction is compatible with the common stacky refinement $\tilde{X}$ defined in previous work (BBB+25), ensuring that the functoriality extends to the common refinement.

Significance
The paper claims that the introduction of the Q-Cartier Cox ring and the associated Q-Cartier stack remedies fundamental deficiencies in the study of nonsimplicial toric varieties. By shifting the focus to Q-Cartier degrees, the authors recover the ideal-variety correspondence and a precise module-sheaf correspondence that were previously unavailable for nonsimplicial cases. Furthermore, the stacky extension of GKZ theory provides a robust framework for studying the birational geometry of toric varieties where canonical stacks fail to preserve functoriality. The results unify the treatment of simplicial and nonsimplicial toric varieties within the language of stacks, offering a "correct" geometric object ($X^{\text{Q-Car}}$) that retains the combinatorial simplicity of the Cox construction while accurately reflecting the geometry of the underlying variety.